let-p-x-x-in-n-n-n-with-n-integr-natural-1-find-the-roots-of-p-x-2-factorize-p-x-inside-C-x-3-decompose-the-fraction-F-x-1-p-x-

Question Number 69795 by mathmax by abdo last updated on 27/Sep/19

l e t p (x) = {(x + i n)}^{n} - n^{n} w i t h n i n t e g r n a t u r a l

1) f i n d t h e r o o t s o f p (x)

2) f a c t o r i z e p (x) i n s i d e C [x]

3) d e c o m p o s e t h e f r a c t i o n F (x) = \frac{1}{p (x)}

Commented by mathmax by abdo last updated on 29/Sep/19

1) p (x) = 0 \Leftrightarrow {(x + i n)}^{n} = n^{n} \Leftrightarrow \frac{{(x + i n)}^{n}}{n^{n}} = 1 \Leftrightarrow {(\frac{x + i n}{n})}^{n} = 1

\Leftrightarrow {(\frac{x}{n} + i)}^{n} = 1 l e t z = \frac{x}{n} + i (e) \Rightarrow z^{n} = 1 \Rightarrow z^{n} = e^{i (2 k π)} \Rightarrow

z_{k} = e^{\frac{i 2 k π}{n}} w i t h k \in [[0, n - 1]] s o w e g e t {n z}_{k} = x_{k} + n i \Rightarrow

x_{k} = n (z_{k} - i) = n (e^{\frac{i 2 k π}{n}} - i) w i t h 0 ⩽ k ⩽ n - 1

2) p (x) = a \prod_{k = 0}^{n - 1} (x - x_{k}) = a \prod_{k = 0}^{n - 1} (x + n i - {n e}^{\frac{i 2 k π}{n}})

w e h a v e p (x) = \sum_{k = 0}^{n} C_{n}^{k} x^{k} {(i n)}^{n - k} - n^{n}

k = n \Rightarrow a = C_{n}^{n} {(i n)}^{0} = 1 \Rightarrow p (x) = \prod_{k = 0}^{n - 1} (x + n i - {n e}^{\frac{i 2 k π}{n}})

3) F (x) = \frac{1}{p (x)} = \frac{1}{\prod_{k = 0}^{n - 1} (x - x_{k})} = \sum_{k = 0}^{n - 1} \frac{λ_{k}}{x - x_{k}}

λ_{k} = \frac{1}{p^{^{'}} (x_{k})}